The MCNP4B Monte Carlo transport code is used in a feasibility study of the epithermal neutron boron neutron capture therapy facility in the thermalizing column of the 250-kW TRIGA Mark II reactor at the Jozef Stefan Institute (JSI). To boost the epithermal neutron flux at the reference irradiation point, the efficiency of a fission plate with almost 1.5 kg of 20% enriched uranium and 2.3 kW of thermal power is investigated. With the same purpose in mind, the TRIGA reactor core setup is optimized, and standard fresh fuel elements are concentrated partly in the outermost ring of the core. Further, a detailed parametric study of the materials and dimensions for all the relevant parts of the irradiation facility is carried out. Some of the standard epithermal neutron filter/moderator materials, as well as "pressed-only" low-density Al2O3 and AlF3, are considered. The proposed version of the BNCT facility, with PbF2 as the epithermal neutron filter/moderator, provides an epithermal neutron flux of ~1.1 × 109 n/cm2s, thus enabling patient irradiation times of <60 min. With reasonably low fast neutron and photon contamination  ([overdot]Dnfast/epi < 5 × 10-13 Gycm2/n and [overdot]D /epi < 3 × 10-13 Gycm2/n), the in-air performances of the proposed beam are comparable to all existing epithermal BNCT facilities. The design presents an equally efficient alternative to the BNCT beams in TRIGA reactor thermal columns that are more commonly applied. The cavity of the dry cell, a former JSI TRIGA reactor spent-fuel storage facility, adjacent to the thermalizing column, could rather easily be rearranged into a suitable patient treatment room, which would substantially decrease the overall developmental costs.